{"title": "An ideal observer model of infant object perception", "book": "Advances in Neural Information Processing Systems", "page_first": 825, "page_last": 832, "abstract": "Before the age of 4 months, infants make inductive inferences about the motions of physical objects. Developmental psychologists have provided verbal accounts of the knowledge that supports these inferences, but often these accounts focus on categorical rather than probabilistic principles. We propose that infant object perception is guided in part by probabilistic principles like persistence: things tend to remain the same, and when they change they do so gradually. To illustrate this idea, we develop an ideal observer model that includes probabilistic formulations of rigidity and inertia. Like previous researchers, we suggest that rigid motions are expected from an early age, but we challenge the previous claim that expectations consistent with inertia are relatively slow to develop (Spelke et al., 1992). We support these arguments by modeling four experiments from the developmental literature.", "full_text": "An ideal observer model of infant object perception\n\nCharles Kemp\n\nDepartment of Psychology\nCarnegie Mellon University\n\nckemp@cmu.edu\n\nFei Xu\n\nDepartment of Psychology\n\nUniversity of British Columbia\n\nfei@psych.ubc.ca\n\nAbstract\n\nBefore the age of 4 months, infants make inductive inferences about the motions\nof physical objects. Developmental psychologists have provided verbal accounts\nof the knowledge that supports these inferences, but often these accounts focus on\ncategorical rather than probabilistic principles. We propose that infant object per-\nception is guided in part by probabilistic principles like persistence: things tend\nto remain the same, and when they change they do so gradually. To illustrate this\nidea we develop an ideal observer model that incorporates probabilistic principles\nof rigidity and inertia. Like previous researchers, we suggest that rigid motions\nare expected from an early age, but we challenge the previous claim that the in-\nertia principle is relatively slow to develop [1]. We support these arguments by\nmodeling several experiments from the developmental literature.\n\nOver the past few decades, ingenious experiments [1, 2] have suggested that infants rely on sys-\ntematic expectations about physical objects when interpreting visual scenes. Looking time studies\nsuggest, for example, that infants expect objects to follow continuous trajectories through time and\nspace, and understand that two objects cannot simultaneously occupy the same location. Many of\nthese studies have been replicated several times, but there is still no consensus about the best way to\ncharacterize the knowledge that gives rise to these \ufb01ndings.\nTwo main approaches can be found in the literature. The verbal approach uses natural language\nto characterize principles of object perception [1, 3]: for example, Spelke [4] proposes that object\nperception is consistent with principles including continuity (\u201ca moving object traces exactly one\nconnected path over space and time\u201d) and cohesion (\u201ca moving object maintains its connectedness\nand boundaries\u201d). The mechanistic approach proposes that physical knowledge is better charac-\nterized by describing the mechanisms that give rise to behavior, and researchers working in this\ntradition often develop computational models that support their theoretical proposals [5]. We pursue\na third approach\u2014the ideal observer approach [6, 7, 8]\u2014that combines aspects of both previous\ntraditions. Like the verbal approach, our primary goal is to characterize principles that account for\ninfant behavior, and we will not attempt to characterize the mechanisms that produce this behavior.\nLike the mechanistic approach, we emphasize the importance of formal models, and suggest that\nthese models can capture forms of knowledge that are dif\ufb01cult for verbal accounts to handle.\nIdeal observer models [6, 9] specify the conclusions that normatively follow given a certain source\nof information and a body of background knowledge. These models can therefore address questions\nabout the information and the knowledge that support perception. Approaches to the information\nquestion characterize the kinds of perceptual information that human observers use. For example,\nGeisler [9] discusses which components of the information available at the retina contribute to vi-\nsual perception, and Banks and Shannon [10] use ideal observer models to study the perceptual\nconsequences of immaturities in the retina. Approaches to the knowledge question characterize the\nbackground assumptions that are combined with the available input in order to make inductive infer-\nences. For example, Weiss and Adelson [7] describe several empirical phenomena that are consistent\nwith the a priori assumption that motions tend to be slow and smooth. There are few previous at-\ntempts to develop ideal observer models of infant perception, and most of them focus only on the\n\n\finformation question [10]. This paper addresses the knowledge question, and proposes that the ideal\nobserver approach can help to identify the minimal set of principles needed to account for the visual\ncompetence of young infants.\nMost verbal theories of object perception focus on categorical principles [4], or principles that make\na single distinction between possible and impossible scenes. We propose that physical knowledge\nin infancy is also characterized by probabilistic principles, or expectations that make some possible\nscenes more surprising than others. We demonstrate the importance of probabilistic principles by\nfocusing on two examples: the rigidity principle states that objects usually maintain their shape and\nsize when they move, and the inertia principle states that objects tend to maintain the same pattern of\nmotion over time. Both principles capture important regularities, but exceptions to these regularities\nare relatively common.\nFocusing on rigidity and inertia allows us to demonstrate two contributions that probabilistic ap-\nproaches can make. First, probabilistic approaches can reinforce current proposals about infant\nperception. Spelke [3] suggests that rigidity is a core principle that guides object perception from a\nvery early age, and we demonstrate how this idea can be captured by a model that also tolerates ex-\nceptions, such as non-rigid biological motion. Second, probabilistic approaches can identify places\nwhere existing proposals may need to be revised. Spelke [3] argues that the principle of inertia is\nslow to develop, but we suggest that a probabilistic version of this principle can help to account for\ninferences made early in development.\n\n1 An ideal observer approach\nAn ideal observer approach to object perception can be formulated in terms of a generative model\nfor scenes. Scenes can be generated in three steps. First we choose the number n of objects that\nwill appear in the scene, and generate the shape, visual appearance, and initial location of each\nobject. We then choose a velocity \ufb01eld for each object which speci\ufb01es how the object moves and\nchanges shape over time. Finally, we create a visual scene by taking a two-dimensional projection\nof the moving objects generated in the two previous steps. An ideal observer approach explores\nthe idea that the inferences made by infants approximate the optimal inferences with respect to this\ngenerative model.\nWe work within this general framework but make two simpli\ufb01cations. We will not discuss how the\nshapes and visual appearances of objects are generated, and we make the projection step simple by\nworking with a two-dimensional world. These simpli\ufb01cations allow us to focus on the expectations\nabout velocity \ufb01elds that guide motion perception in infants. The next two sections present two prior\ndistributions that can be used to generate velocity \ufb01elds. The \ufb01rst is a baseline prior that does not\nincorporate probabilistic principles, and the second incorporates probabilistic versions of rigidity\nand inertia. The two priors capture different kinds of knowledge, and we argue that the second\nprovides the more accurate characterization of the knowledge that infants bring to object perception.\n\n1.1 A baseline prior on velocity \ufb01elds\nOur baseline prior is founded on \ufb01ve categorical principles that are closely related to principles\ndiscussed by Spelke [3, 4]. The principles we consider rely on three basic notions: space, time, and\nmatter. We also refer to particles, which are small pieces of matter that occupy space-time points.\nParticles satisfy several principles:\n\nC1. Temporal continuity. Particles are not created or destroyed. In other words, every particle\n\nthat exists at time t1 must also exist at time t2.\n\nC2. Spatial continuity. Each particle traces a continuous trajectory through space.\nC3. Exclusion. No two particles may occupy the same space-time point.\n\nAn object is a collection of particles, and these collections satisfy two principles:\n\nC4. Discreteness. Each particle belongs to exactly one object.\nC5. Cohesion. At each point in time, the particles belonging to an object occupy a single\n\nconnected region of space.\n\nSuppose that we are interested in a space-time window speci\ufb01ed by a bounded region of space and a\nbounded interval of time. For simplicity, we will assume that space is two-dimensional, and that the\nspace-time window corresponds to the unit cube. Suppose that a velocity \ufb01eld ~v assigns a velocity\n\n\f(vx, vy) to each particle in the space-time window, and let ~vi be the \ufb01eld created by considering\nonly particles that belong to object i. We develop a theory of object perception by de\ufb01ning a prior\ndistribution p(~v) on velocity \ufb01elds.\nConsider \ufb01rst the distribution p( ~v1) on \ufb01elds for a single object. Any \ufb01eld that violates one or more\nof principles C1\u2013C5 is assigned zero probability. For instance, \ufb01elds where part of an object winks\nout of existence violate the principle of temporal continuity, and \ufb01elds where an object splits into\ntwo distinct pieces violate the principle of cohesion. Many \ufb01elds, however, remain, including \ufb01elds\nthat specify non-rigid motions and jagged trajectories. For now, assume that we are working with\na space of \ufb01elds that is bounded but very large, and that the prior distribution over this space is\nuniform for all \ufb01elds consistent with principles C1\u2013C5:\n\np( ~v1) \u221d f ( ~v1) = (cid:26) 0\n\n1\n\nif ~v1 violates C1\u2013C5\notherwise.\n\n(1)\n\nConsider now the distribution p( ~v1, ~v2) on \ufb01elds for pairs of objects. Principles C1 through C5 rule\nout some of these \ufb01elds, but again we must specify a prior distribution on those that remain. Our\nprior is induced by the following principle:\n\nC6. Independence. Velocity \ufb01elds for multiple objects are independently generated subject to\n\nprinciples C1 through C5.\n\nf ( ~v1) . . . f ( ~vn) otherwise.\n\n(2)\n\nif {~vi} collectively violate C1\u2013C5\n\nMore formally, the independence principle speci\ufb01es how the prior for the multiple object case is\nrelated to the prior p( ~v1) on velocity \ufb01elds for a single object (Equation 1):\np( ~v1, . . . , ~vn) \u221d f ( ~v1, . . . , ~vn) = (cid:26) 0\n1.2 A smoothness prior on velocity \ufb01elds\nWe now develop a prior p(~v) that incorporates probabilistic expectations about the motion of phys-\nical objects. Consider again the prior p( ~v1) on the velocity \ufb01eld ~v1 of a single object. Principles\nC1\u2013C5 make a single cut that distinguishes possible from impossible \ufb01elds, but we need to consider\nwhether infants have additional knowledge that makes some of the possible \ufb01elds less surprising\nthan others. One informal idea that seems relevant is the notion of persistence[11]: things tend to\nremain the same, and when they change they do so gradually. We focus on two versions of this idea\nthat may guide expectations about velocity \ufb01elds:\n\nS1. Spatial smoothness. Velocity \ufb01elds tend to be smooth in space.\nS2. Temporal smoothness. Velocity \ufb01elds tend to be smooth in time.\n\nA \ufb01eld is \u201csmooth in space\u201d if neighboring particles tend to have similar velocities at any instant\nof time. The smoothest possible \ufb01eld will be one where all particles have the same velocity at\nany instant\u2014in other words, where an object moves rigidly. The principle of spatial smoothness\ntherefore captures the idea that objects tend to maintain the same shape and size.\nA \ufb01eld is \u201csmooth in time\u201d if any particle tends to have similar velocities at nearby instants of time.\nThe smoothest possible \ufb01eld will be one where each particle maintains the same velocity throughout\nthe entire interval of interest. The principle of temporal smoothness therefore captures the idea that\nobjects tend to maintain their initial pattern of motion. For instance, stationary objects tend to remain\nstationary, moving objects tend to keep moving, and a moving object following a given trajectory\ntends to continue along that trajectory.\nPrinciples S1 and S2 are related to two principles\u2014 rigidity and inertia\u2014that have been discussed\nin the developmental literature. The rigidity principle states that objects \u201ctend to maintain their size\nand shape over motion\u201d[3], and the inertia principle states that objects move smoothly in the absence\nof obstacles [4]. Some authors treat these principles rather differently: for instance, Spelke suggests\nthat rigidity is one of the core principles that guides object perception from a very early age [3], but\nthat the principle of inertia is slow to develop and is weak or fragile once acquired. Since principles\nS1 and S2 seem closely related, the suggestion that one develops much later than the other seems\ncounterintuitive. The rest of this paper explores the idea that both of these principles are needed to\ncharacterize infant perception.\nOur arguments will be supported by formal analyses, and we therefore need formal versions of\nS1 and S2. There may be different ways to formalize these principles, but we present a simple\n\n\fa)\n\nL1\n\nL2\n\nU\n\nb)\n\n200\n\n0\n\n\u201d\n)\n)\n~v\n~v\n|\n|\n1\n2\nH\nH\n(\n(\np\np\n\u201c\n\nL1\n\nL2\n\nU\n\ng\no\nl\n\n\u2212200\n\nFigure 1: (a) Three scenes inspired by the experiments of Spelke and colleagues [12, 13]. Each\nscene can be interpreted as a single object, or as a small object on top of a larger object. (b) Relative\npreferences for the one-object and two-object interpretations according to two models. The baseline\nmodel prefers the one-object interpretation in all three cases, but the smoothness model prefers the\none-object interpretation only for scenes L1 and L2.\n\nbaseline\n\nsmoothness\n\napproach that builds on existing models of motion perception in adults [7, 8]. We de\ufb01ne measures\nof instantaneous roughness that capture how rapidly a velocity \ufb01eld ~v varies in space and time:\n\nRspace(~v, t) =\n\nRtime(~v, t) =\n\n1\n\nvol(O(t)) Z\n\nO(t)\n\n1\n\nvol(O(t)) Z\n\nO(t)\n\n\u2202~v(x, y, t)\n\n\u2202x\n\n\u2202~v(x, y, t)\n\n\u2202t\n\n(cid:12)(cid:12)(cid:12)(cid:12)\n(cid:12)(cid:12)(cid:12)(cid:12)\n\n(cid:12)(cid:12)(cid:12)(cid:12)\n(cid:12)(cid:12)(cid:12)(cid:12)\n\n2\n\n2\n\n+(cid:12)(cid:12)(cid:12)(cid:12)\n\n\u2202~v(x, y, t)\n\n\u2202y\n\ndxdy\n\n2\n\n(cid:12)(cid:12)(cid:12)(cid:12)\n\ndxdy\n\n(3)\n\n(4)\n\nwhere O(t) is the set of all points that are occupied by the object at time t, and vol(O(t)) is the\nvolume of the object at time t. Rspace(~v, t) will be large if neighboring particles at time t tend to\nhave different velocities, and Rtime(~v, t) will be large if many particles are accelerating at time t.\nWe combine our two roughness measures to create a single smoothness function S(\u00b7) that measures\nthe smoothness of a velocity \ufb01eld:\n\nS(~v) = \u2212\u03bbspaceZ Rspace(~v, t)dt \u2212 \u03bbtimeZ Rtime(~v, t)dt\n\n(5)\nwhere \u03bbspace and \u03bbtime are positive weights that capture the importance of spatial smoothness and\ntemporal smoothness. For all analyses in this paper we set \u03bbspace = 10000 and \u03bbtime = 250, which\nimplies that violations of spatial smoothness are penalized more harshly than violations of temporal\nsmoothness. We now replace Equation 1 with a prior on velocity \ufb01elds that takes smoothness into\naccount:\n\np( ~v1) \u221d f ( ~v1) = (cid:26) 0\n\nexp (S( ~v1)) otherwise.\n\nif ~v1 violates C1\u2013C5\n\n(6)\n\nCombining Equation 6 with Equation 2 speci\ufb01es a model of object perception that incorporates\nprobabilistic principles of rigidity and inertia.\n2 Empirical \ufb01ndings: spatial smoothness\nThere are many experiments where infants aged 4 months and younger appear to make inferences\nthat are consistent with the principle of rigidity. This section suggests that the principle of spatial\nsmoothness can account for these results. We therefore propose that a probabilistic principle (spatial\nsmoothness) can explain all of the \ufb01ndings previously presented in support of a categorical principle\n(rigidity), and can help in addition to explain how infants perceive non-rigid motion.\nOne set of studies explores inferences about the number of objects in a scene. When a smaller block\nis resting on top of a larger block (L1 in Figure 1a), 3-month-olds infer that the scene includes a\nsingle object [12]. The same result holds when the small and large blocks are both moving in the\nsame direction (L2 in Figure 1a) [13]. When these blocks are moving in opposite directions (U in\nFigure 1a), however, infants appear to infer that the scene contains two objects [13]. Results like\nthese suggest that infants may have a default expectation that objects tend to move rigidly.\nWe compared the predictions made by two models about the scenes in Figure 1a. The smoothness\nmodel uses a prior p( ~v1) that incorporates principles S1 and S2 (Equation 6), and the baseline model\nis identical except that it sets \u03bbspace = \u03bbtime = 0. Both models therefore incorporate principles C1\u2013\nC6, but only the smoothness model captures the principle of spatial smoothness.\n\n\fGiven any of the scenes in Figure 1a, an infant must solve two problems: she must compute the\nvelocity \ufb01eld ~v for the scene and must decide whether this \ufb01eld speci\ufb01es the motion of one or two\nobjects. Here we focus on the second problem, and assume that the infant\u2019s perceptual system has\nalready computed a veridical velocity \ufb01eld for each scene that we consider. In principle, however,\nthe smoothness prior in Equation 6 can address both problems. Previous authors have shown how\nsmoothness priors can be used to compute velocity \ufb01elds given raw image data [7, 8].\nLet H1 be the hypothesis that a given velocity \ufb01eld corresponds to a single object, and let H2 be the\nhypothesis that the \ufb01eld speci\ufb01es the motions of two objects. We assume that the prior probabilities\nof these hypotheses are equal, and that P (H1) = P (H2) = 0.5. An ideal observer can use the\nposterior odds ratio to choose between these hypotheses:\n\nP (H1|~v)\nP (H2|~v)\n\n=\n\nP (~v|H1)\nP (~v|H2)\n\nP (H1)\nP (H2)\n\n\u2248\n\nf (~v)\n\nR f ( ~v1)d ~v1 R f ( ~v1, ~v2)d ~v1d ~v2\n\nf ( ~vA, ~vB)\n\n(7)\n\nEquation 7 follows from Equations 2 and 6, and from approximating P (~v|H2) by considering only\nthe two object interpretation ( ~vA, ~vB) with maximum posterior probability. For each scene in Fig-\nure 1a, the best two object interpretation will specify a \ufb01eld ~vA for the small upper block, and a \ufb01eld\n~vB for the large lower block.\nTo approximate the posterior odds ratio in Equation 7 we compute rough approximations of\nR f ( ~v1)d ~v1 andR f ( ~v1, ~v2)d ~v1d ~v2 by summing over a \ufb01nite space of velocity \ufb01elds. As described in\nthe supporting material, we consider all \ufb01elds that can be built from objects with 5 possible shapes,\n900 possible starting locations, and 10 possible trajectories. For computational tractability, we con-\nvert each continuous velocity \ufb01eld to a discrete \ufb01eld de\ufb01ned over a space-time grid with 45 cells\nalong each spatial dimension and 21 cells along the temporal dimension.\nOur results show that both models prefer the one-object hypothesis H1 when presented with scenes\nL1 and L2 (Figure 1b). Since there are many more two-object scenes than one-object scenes, any\ntypical two-object interpretation is assigned lower prior probability than a typical one-object inter-\npretation. This preference for simpler interpretations is a consequence of the Bayesian Occam\u2019s\nrazor. The baseline model makes the same kind of inference about scene U, and again prefers the\none-object interpretation. Like infants, however, the smoothness model prefers the two-object in-\nterpretation of scene U. This model assigns low probability to a one-object interpretation where\nadjacent points on the object have very different velocities, and this preference for smooth motion\nis strong enough to overcome the simplicity preference that makes the difference when interpreting\nthe other two scenes.\nOther experiments from the developmental literature have produced results consistent with the prin-\nciple of spatial smoothness. For example, 3.5-month olds are surprised when a tall object is fully\nhidden behind a short screen, 4 month olds are surprised when a large object appears to pass through\na small slot, and 4.5-month olds expect a swinging screen to be interrupted when an object is placed\nin its path [1, 2]. All three inferences appear to rely on the expectation that objects tend not to shrink\nor to compress like foam rubber. Many of these experiments are consistent with an account that\nsimply rules out non-rigid motion instead of introducing a graded preference for spatial smoothness.\nBiological motions, however, are typically non-rigid, and experiments suggest that infants can track\nand make inferences about objects that follow non-rigid trajectories [14]. Findings like these call\nfor a theory like ours that incorporates a preference for rigid motion, but recognizes that non-rigid\nmotions are possible.\n\n3 Empirical \ufb01ndings: temporal smoothness\nWe now turn to the principle of temporal smoothness (S2) and discuss some of the experimental\nevidence that bears on this principle. Some researchers suggest that a closely related principle\n(inertia) is slow to develop, but we argue that expectations about temporal smoothness are needed to\ncapture inferences made before the age of 4 months.\nBaillargeon and DeVos [15] describe one relevant experiment that explores inferences about moving\nobjects and obstacles. During habituation, 3.5-month-old infants saw a car pass behind an occluder\nand emerge from the other side (habituation stimulus H in Figure 2a). An obstacle was then placed\nin the direct path of the car (unlikely scenes U1 and U2) or beside this direct path (likely scene L),\nand the infants again saw the car pass behind the occluder and emerge from the other side. Looking\n\n\fa)\n\nH\n\nL\n\nU1\n\nU2\n\nb)\n\n600\n400\n200\n0\n\np(U 1)\u201d\nlog \u201c p(L)\nlog \u201c p(L)\np(U 2)\u201d\nlog \u201c pH (L)\npH (U 1)\u201d\nlog \u201c pH (L)\npH (U 2)\u201d\n\nX\n\nX\nX\nbaseline\n\nsmoothness\n\nFigure 2: (a) Stimuli inspired by the experiments of [15]. The habituation stimulus H shows a block\npassing behind a barrier and emerging on the other side. After habituation, a new block is added\neither out of the direct path of the \ufb01rst block (L) or directly in the path of the \ufb01rst block (U1 and\nU2). In U1, the \ufb01rst block leaps over the second block, and in U2 the second block hops so that\nthe \ufb01rst block can pass underneath. (b) Relative probabilities of scenes L, U1 and U2 according to\ntwo models. The baseline model \ufb01nds all three scenes equally likely a priori, and considers L and\nU2 equally likely after habituation. The smoothness model considers L more likely than the other\nscenes both before and after habituation.\n\na)\n\nH1\n\nH2\n\nL\n\nU\n\nb)\n\nc)\n\n300\n200\n100\n0\n\u2212100\n\np(U )\u201d\nlog \u201c p(L)\nlog \u201c pH 1(L)\npH 1(U )\u201d\nlog \u201c pH 2(L)\npH 2(U )\u201d\n\nX\n\nX\n\nbaseline\n\nsmoothness\n\nFigure 3: (a) Stimuli inspired by the experiments of Spelke et al. [16]. (b) Model predictions. After\nhabituation to H1, the smoothness model assigns roughly equal probabilities to L and U. After\nhabituation to H2, the model considers L more likely. (c) A stronger test of the inertia principle.\nNow the best interpretation of stimulus U involves multiple changes of direction.\n\ntime measurements suggested that the infants were more surprised to see the car emerge when the\nobstacle lay within the direct path of the car. This result is consistent with the principle of temporal\nsmoothness, which suggests that infants expected the car to maintain a straight-line trajectory, and\nthe obstacle to remain stationary.\nWe compared the smoothness model and the baseline model on a schematic version of this task. To\nmodel this experiment, we again assume that the infant\u2019s perceptual system has recovered a veridical\nvelocity \ufb01eld, but now we must allow for occlusion. An ideal observer approach that treats a two\ndimensional scene as a projection of a three dimensional world can represent the occluder as an\nobject in its own right. Here, however, we continue to work with a two dimensional world, and treat\nthe occluded parts of the scene as missing data. An ideal observer approach should integrate over all\npossible values of the missing data, but for computational simplicity we approximate this approach\nby considering only one or two high-probability interpretations of each occluded scene.\nWe also need to account for habituation, and for cases where the habituation stimulus includes oc-\nclusion. We assume that an ideal observer computes a habituation \ufb01eld ~vH, or the velocity \ufb01eld with\nmaximum posterior probability given the habituation stimulus. In Figure 2a, the inferred habituation\n\ufb01eld ~vH speci\ufb01es a trajectory where the block moves smoothly from the left to the right of the scene.\nWe now assume that the observer expects subsequent velocity \ufb01elds to be similar to ~vH. Formally,\nwe use a product-of-experts approach to de\ufb01ne a post-habituation distribution on velocity \ufb01elds:\n\npH (~v) \u221d p(~v)p(~v| ~vH )\n\n(8)\nThe \ufb01rst expert p(~v) uses the prior distribution in Equation 6, and the second expert p(~v| ~vH ) assumes\nthat \ufb01eld ~v is drawn from a Gaussian distribution centered on ~vH. Intuitively, after habituation to ~vH\nthe second expert expects that subsequent velocity \ufb01elds will be similar to ~vH. More information\nabout this model of habituation is provided in the supporting material.\nGiven these assumptions, the black and dark gray bars in Figure 2 indicate relative a priori proba-\nbilities for scenes L, U1 and U2. The baseline model considers all three scenes equally probable,\n\n\fbut the smoothness model prefers L. After habituation, the baseline model is still unable to account\nfor the behavioral data, since it considers scenes L and U2 to be equally probable. The smoothness\nmodel, however, continues to prefer L.\nWe previously mentioned three consequences of the principle of temporal smoothness: stationary\nobjects tend to remain stationary, moving objects tend to keep moving, and moving objects tend\nto maintain a steady trajectory. The \u201ccar and obstacle\u201d task addresses the \ufb01rst and third of these\nproposals, but other tasks provide support for the second. Many authors have studied settings where\none moving object comes to a stop, and a second object starts to move [17]. Compared to the case\nwhere the \ufb01rst object collides with the second, infants appear to be surprised by the \u201cno-contact\u201d\ncase where the two objects never touch. This \ufb01nding is consistent with the temporal smoothness\nprinciple, which predicts that infants expect the \ufb01rst object to continue moving until forced to stop,\nand expect the second object to remain stationary until forced to start.\nOther experiments [18] provide support for the principle of temporal smoothness, but there are also\nstudies that appear inconsistent with this principle. In one of these studies [16], infants are initially\nhabituated to a block that moves from one corner of an enclosure to another (H1 in Figure 3a).\nAfter habituation, infants see a block that begins from a different corner, and now the occluder\nis removed to reveal the block in a location consistent with a straight-line trajectory (L) or in a\nlocation that matches the \ufb01nal resting place during the habituation phase (U). Looking times suggest\nthat infants aged 4-12 months are no more surprised by the inertia-violating outcome (U) than the\ninertia-consistent outcome (L). The smoothness model, however, can account for this \ufb01nding. The\noutcome in U is contrary to temporal smoothness but consistent with habituation, and the tradeoff\nbetween these factors leads the model to assign roughly the same probability to scenes L and U\n(Figure 3b).\nOnly one of the inertia experiments described by Spelke et al. [16] and Spelke et al. [1] avoids this\ntradeoff between habituation and smoothness. This experiment considers a case where the habitua-\ntion stimulus (H2 in Figure 3a) is equally similar to the two test stimuli. The results suggest that 8\nmonth olds are now surprised by the inertia-violating outcome, and the predictions of our model are\nconsistent with this \ufb01nding (Figure 3b). 4 and 6 month olds, however, continue to look equally at the\ntwo outcomes. Note, however, that the trajectories in Figure 3 include at most one in\ufb02ection point.\nExperiments that consider trajectories with many in\ufb02ection points can provide a more powerful way\nof exploring whether 4 month olds have expectations about temporal smoothness.\nOne possible experiment is sketched in Figure 3c. The task is very similar to the task in Figure 3a,\nexcept that a barrier is added after habituation. In order for the block to end up in the same location\nas before, it must now follow a tortuous path around the barrier (U). Based on the principle of\ntemporal smoothness, we predict that 4-month-olds will be more surprised to see the outcome in\nstimulus U than the outcome in stimulus L. This experimental design is appealing in part because\nprevious work shows that infants are surprised by a case similar to U where the barrier extends all\nthe way from one wall to the other [16], and our proposed experiment is a minor variant of this task.\nAlthough there is room for debate about the status of temporal smoothness, we presented two rea-\nsons to revisit the conclusion that this principle develops relatively late. First, some version of this\nprinciple seems necessary to account for experiments like the car and obstacle experiment in Fig-\nure 2. Second, most of the inertia experiments that produced null results use a habituation stimulus\nwhich may have prevented infants from revealing their default expectations, and the one experiment\nthat escapes this objection considers a relatively minor violation of temporal smoothness. Additional\nexperiments are needed to explore this principle, but we predict that the inertia principle will turn\nout to be yet another example of knowledge that is available earlier than researchers once thought.\n\n4 Discussion and Conclusion\nWe argued that characterizations of infant knowledge should include room for probabilistic expecta-\ntions, and that probabilistic expectations about spatial and temporal smoothness appear to play a role\nin infant object perception. To support these claims we described an ideal observer model that in-\ncludes both categorical (C1 through C5) and probabilistic principles (S1 and S2), and demonstrated\nthat the categorical principles alone are insuf\ufb01cient to account for several experimental \ufb01ndings. Our\ntwo probabilistic principles are related to principles (rigidity and inertia) that have previously been\ndescribed as categorical principles. Although rigidity and inertia appear to play a role in some early\n\n\finferences, formulating these principles as probabilistic expectations helps to explain how infants\ndeal with non-rigid motion and violations of inertia.\nOur analysis focused on some of the many existing experiments in the developmental literature, but\nnew experiments will be needed to explore our probabilistic approach in depth. Categorical versions\nof a given principle (e.g. rigidity) allow room for only two kinds of behavior depending on whether\nthe principle is violated or not. Probabilistic principles can be violated to a greater or lesser extent,\nand our approach predicts that violations of different magnitude may lead to different behaviors.\nFuture studies of rigidity and inertia can consider violations of these principles that range from\nmild (Figure 3a) to severe (Figure 3c), and can explore whether infants respond to these violations\ndifferently. Future work should also consider whether the categorical principles we described (C1\nthrough C5) are better characterized as probabilistic expectations. In particular, future studies can\nexplore whether young infants consider large violations of cohesion (C5) or spatial continuity (C2)\nmore surprising than smaller violations of these principles.\nAlthough we did not focus on learning, our approach allows us to begin thinking formally about\nhow principles of object perception might be acquired. First, we can explore how parameters like\nthe smoothness parameters in our model (\u03bbspace and \u03bbtime) might be tuned by experience. Second,\nwe can use statistical model selection to explore transitions between different sets of principles.\nFor instance, if a learner begins with the baseline model we considered (principles C1\u2013C6), we\ncan explore which subsequent observations provide the strongest statistical evidence for smoothness\nprinciples S1 and S2, and how much of this evidence is required before an ideal learner would\nprefer our smoothness model over the baseline model. It is not yet clear which principles of object\nperception could be learned, but the ideal observer approach can help to resolve this question.\nReferences\n[1] E. S. Spelke, K. Breinlinger, J. Macomber, and K. Jacobson. Origins of knowledge. Psychological Review,\n\n99:605\u2013632, 1992.\n\n[2] R. Baillargeon, L. Kotovsky, and A. Needham. The acquisition of physical knowledge in infancy.\n\nIn\nD. Sperber, D. Premack, and A. J. Premack, editors, Causal Cognition: A multidisciplinary debate, pages\n79\u2013116. 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